1
曆五○庚午元曆上
Calendar 5 — Gengwu Origin Calendar, Part One.
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演紀上元庚午，距太祖庚辰歲，積年二千二十七萬五千二百七十算外，上考往古，每年減一算，下驗將來，每年加一算。 步氣朔術
The epoch reckons the high origin at gengwu, at an interval from Taizu's gengchen year of 20,275,270 years beyond the count; to examine antiquity upward, subtract one count per year; to verify the future downward, add one count per year. Stepwise computation: qi and lunation.
3
日法，五千二百三十。 歲實，一百九十一萬二百二十四。 通余，二萬七千四百二十四。 朔實，一十五萬四千四百四十五。
Day unit: 5,230. Tropical year length: 1,910,024 parts. Universal remainder: 27,424 parts. Synodic-month constant: 154,445 parts.
4
通閏，五萬六千八百八十四。 歲策，三百六十五，余一千二百七十四。 朔策，二十九，余二千七百七十五。 氣策，一十五，余一千一百四十二，秒六十。
Universal intercalation: 56,884 parts. Year divisor: 365 days, remainder 1,274 parts. Synodic-month divisor: 29 days, remainder 2,775 parts. Qi divisor: 15 days, remainder 1,142 parts 60 seconds.
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望策，一十四，余四千二，秒四十五。 象策，七，余二千一，秒二十二半。 沒限，四千八十七，秒三十。 朔虛分，二千四百五十五。
Full-moon divisor: 14 days, remainder 4,002 parts 45 seconds. Quarter-moon divisor: 7 days, remainder 2,001 parts 22½ seconds. Extinction limit: 4,087 parts 30 seconds. New-moon void parts: 2,455.
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旬周，三十一萬三千八百。 紀法，六十。 秒母，九十。 求天正冬至
Ten-day cycle: 313,800 parts. Era divisor: 60. Second divisor: 90. To find the winter solstice of the civil year.
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置上元庚午以來積年，以歲實乘之，為通積分； 滿旬周，去之，不盡，以日法約之，為日，不盈，為余； 命壬戌算外，即得所求天正冬至大小余也。 〈（先以裏差加減通積分，然後求之。 求裏差術，具《月離》篇中。）〉
Set the accumulated years since the gengwu high origin, multiply by the tropical year length, and obtain the universal accumulated parts; divide by the ten-day cycle and discard the quotient; with the remainder, reduce by the day unit to obtain days, and what does not fill a day becomes the remainder; count from renxu beyond the counting rod, and the major and minor remainders of the sought civil-year winter solstice are obtained. (First apply the inner difference to add or subtract from the universal accumulated parts, then proceed with the calculation. (The method for finding the inner difference is given in full in the 《Lunar Motion》 section.)
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求次氣
To find the next solar term.
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置天正冬至大小余，以氣策及余累加之，秒盈秒母從分，分滿日法從日，即得次氣日及余分秒。 求天正經朔
Set the major and minor remainders of the civil-year winter solstice and add the qi divisor and its remainder cumulatively; when seconds fill the second divisor, carry to parts, and when parts fill the day unit, carry to days; the day and remainder in parts and seconds of the next solar term are thereby obtained. To find the mean new moon of the civil year.
10
置通積分，滿朔實去之，不盡，為閏余； 以減通積分，為朔積分； 滿旬周，去之，不盡，如日法而一，為日，不盡，為余，即得所求天正經朔大小余也。
Set the universal accumulated parts, divide by the synodic-month constant and take the remainder as the intercalation remainder; subtract this from the universal accumulated parts to obtain the new-moon accumulated parts; divide by the ten-day cycle and discard the quotient; with the remainder, divide by the day unit to obtain days, and what remains becomes the fractional remainder; the major and minor remainders of the sought civil-year mean new moon are thereby obtained.
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求弦望及次朔
To find the quarter and full moons and the next new moon.
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置天正經朔大小余，以象策累加之，即各得弦望及次朔經日及余秒也。 求沒日
Set the major and minor remainders of the civil-year mean new moon and add the quarter-moon divisor cumulatively; the mean days and remainders in seconds for each quarter moon, full moon, and subsequent new moon are thereby obtained. To find the extinction day.
13
置有沒之氣恒氣小余，如沒限以上，為有沒之氣； 以秒母乘之，內其秒，用減四十七萬七千五百五十六； 余，滿六千八百五十六而一； 所得並入恒氣大余內，命壬戌算外，即得為沒日也。
Set the minor remainder of the constant qi for a term that bears an extinction; if it reaches the extinction limit or above, that term has an extinction; multiply by the second divisor and include the seconds, then subtract the result from 477,556; with the remainder, divide by 6,856; add the quotient to the major remainder of the constant qi, count from renxu beyond the counting rod, and the extinction day is obtained.
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求滅日
To find the annihilation day.
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置有滅之朔小余， 〈（經朔小余不滿朔虛分者。）〉 六因之，如四百九十一而一； 所得並經朔大余，命為滅日。 步卦候發斂術候策，五，余三百八十，秒八十。
Set the minor remainder of the new moon that bears an annihilation, (when the mean new moon's minor remainder does not fill the new-moon void parts.) multiply by six and divide by 491; add the quotient to the mean new moon's major remainder to obtain the annihilation day. Stepwise computation: hexagrams, hou, and issuing-and-gathering. Hou divisor: 5, remainder 380 parts 80 seconds.
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卦策，六，余四百五十七，秒六。 貞策，三，余二百二十八，秒四十八。 秒母，九十。 辰法，二千六百一十五。
Hexagram divisor: 6, remainder 457 parts 6 seconds. Zhen divisor: 3, remainder 228 parts 48 seconds. Second divisor: 90. Double-hour divisor: 2,615.
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半辰法，一千三百七半。 刻法，三百一十三，秒八十。 辰刻，八，分一百四，秒六十。 半辰刻，四，分五十二，秒三十。
Half double-hour divisor: 1,307½. Quarter-hour divisor: 313 parts 80 seconds. Double-hour quarter-hours: 8, 104 parts 60 seconds. Half double-hour quarter-hours: 4, 52 parts 30 seconds.
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秒母，一百。 求七十二候
Second divisor: 100. To find the seventy-two hou.
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置節氣大小余，命之為初候； 以候策累加之，即得次候及末候也。 求六十四卦
Set the major and minor remainders of the solar term and designate them as the first hou; add the hou divisor cumulatively to obtain the second hou and the final hou. To find the sixty-four hexagrams.
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置中氣大小余，命之為公卦； 以卦策累加之，得辟卦； 又加，得候內卦； 以貞策加之，得節氣之初，為候外卦； 又以貞策加之，得大夫卦； 又以卦策加之，為卿卦也。
Set the major and minor remainders of the mid-term and designate them as the duke hexagram; add the hexagram divisor cumulatively to obtain the lord hexagram; add again to obtain the inner hou hexagram; add the zhen divisor to reach the beginning of the solar term, which is the outer hou hexagram; add the zhen divisor again to obtain the great officer hexagram; add the hexagram divisor again to obtain the minister hexagram.
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求土王用事以貞策減四季中氣大小余，即得土王用事日也。 求發斂
To find the day when Earth holds royal sway: subtract the zhen divisor from the major and minor remainders of the mid-terms of the four seasons, and the day of Earth's royal sway is obtained. To find the issuing and gathering.
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置小余，以六因之，如辰法而一，為辰數； 不盡，以刻法除為刻，命子正算外，即得加時所在辰刻分也。 〈（如加半辰法，即命子初。）〉 求二十四氣卦候
Set the minor remainder, multiply by six, and divide by the double-hour divisor to obtain the double-hour count; with the remainder, divide by the quarter-hour divisor to obtain quarter-hours; count from midnight beyond the counting rod, and the double-hour, quarter-hour, and parts of the time-of-day are obtained. (If the half double-hour divisor is added, count from the beginning of zi.) To find the hexagrams and hou for the twenty-four solar terms.
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〈（以下表格略）〉 步日躔術周天分，一百九十一萬二百九十二，秒九十八。 歲差，六十八，秒九十八。
(The tables below are omitted.) Stepwise computation: solar motion. Circuit-of-heaven parts: 1,910,092 parts 98 seconds. Annual precession: 68 parts 98 seconds.
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秒母，一百。 周天度，三百六十五，分二十五，秒六十七。 象限，九十一，分三十一，秒九。 分秒母，一百。
Second divisor: 100. Circuit of heaven: 365 degrees, 25 parts 67 seconds. Quadrant limit: 91 degrees 31 parts 9 seconds. Parts-and-seconds divisor: 100.
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二十四氣日積度盈縮 〈（表略）〉 二十四氣中積及朓朒 〈（表略）〉
Daily accumulated solar degrees: expansion and contraction for the twenty-four qi. (Table omitted.) Mid-accumulation and lunar wanings and waxings for the twenty-four qi. (Table omitted.)
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求每日盈縮朓朒
To find the daily expansion and contraction and lunar waning and waxing.
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各置其氣損益率， 〈（求盈縮，用盈縮之損益； 求朓朒，用朓朒之損益。）〉 六因，如象限而一，為其氣中率； 與後氣中率相減，為合差； 半合差，加減其氣中率，為元末泛率， 〈（至後，加初減末； 分後，減初加末。）〉 又置合差，六因，如象限而一，為日差； 半之，加減初末泛率，為初末定率； 〈（至後，減初加末； 分後，加初減末。）〉 以日差累加減氣初定率，為每日損益分； 〈（至後，減； 分後，加。）〉 各以每日損益分加減氣下盈縮朓朒，為每日盈縮朓朒。 〈（二分前一氣無後率相減為合差者，皆用前氣合差。）〉
For each term, set its increase-decrease rate, (To find expansion and contraction, use the increase-decrease of expansion and contraction; (to find waning and waxing, use the increase-decrease of waning and waxing.) multiply by six and divide by the quadrant limit to obtain the mid-rate for that term; subtract it from the next term's mid-rate to obtain the combined difference; take half the combined difference and add or subtract it from the term's mid-rate to obtain the general rate at the origin and end, (After a solstice, add at the beginning and subtract at the end; (after an equinox, subtract at the beginning and add at the end.) again set the combined difference, multiply by six, divide by the quadrant limit, and obtain the daily difference; halve it and add or subtract from the general rates at beginning and end to obtain the fixed rates at beginning and end; (After a solstice, subtract at the beginning and add at the end; (after an equinox, add at the beginning and subtract at the end.) add or subtract the daily difference cumulatively from the fixed rate at the term's beginning to obtain each day's increase-decrease parts; (After a solstice, subtract; (after an equinox, add.) for each day, add or subtract the daily increase-decrease parts from the term's expansion-contraction and waning-waxing values to obtain the daily expansion-contraction and waning-waxing. (For the term before each equinox that lacks a following rate to form the combined difference, use the previous term's combined difference instead.)
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求經朔弦望入氣
To find the solar term entered by the mean new moon, quarter moon, and full moon.
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置天正閏余，以日法除為日，不滿，為余。 如氣策以下，以減氣策，為入大雪氣； 以上，去之，余亦以減氣策，為入小雪氣； 即得天正經朔入氣日及余也。 以象策累加之，滿氣策去之，即為弦望入次氣日及余； 因加得後朔入氣日及余也。 〈（便為中朔望入氣。）〉
Set the civil-year intercalation remainder, divide by the day unit to obtain days, and what does not fill a day becomes the remainder. If it reaches the qi divisor or below, subtract it from the qi divisor to obtain entry into the Great Snow term; if above, discard one qi divisor; with the remainder also subtract the qi divisor to obtain entry into the Lesser Snow term; the day and remainder of the civil-year mean new moon's entry into a solar term are thereby obtained. Add the quarter-moon divisor cumulatively and, when it fills the qi divisor, discard the quotient to obtain the day and remainder of entry into the next term for the quarter and full moons; by the same addition the day and remainder of the next new moon's entry into a term are obtained. (This directly gives the mid-term entry for new and full moons.)
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求每日損益盈縮朓朒
To find the daily increase-decrease, expansion-contraction, and waning-waxing.
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以日差益加損減其氣初損益率，為每日損益率； 馴積損益其氣盈縮朓朒積，為每日盈縮朓朒積。 求經朔弦望入氣朓朒定數
Add or subtract the daily difference from the term's initial increase-decrease rate to obtain the daily increase-decrease rate; accumulate in sequence the increase-decrease applied to the term's expansion-contraction and waning-waxing totals to obtain the daily expansion-contraction and waning-waxing accumulations. To find the fixed waning-waxing number for the mean new moon, quarter moon, and full moon at their entry into a term.
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以各所求入氣小余，以乘其日損益率，如日法而一； 所得，損益其下朓朒積，為定數。 〈（便為中朔弦望朓朒定數。）〉 赤道宿度
Multiply each sought entry-into-qi minor remainder by that day's increase-decrease rate and divide by the day unit; add or subtract the result from the waning-waxing accumulation below to obtain the fixed number. (This directly gives the fixed waning-waxing number for the mid-term new moon, quarter moon, and full moon.) Degrees of the equatorial lodges.
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鬥二十五牛七 （少） 女十一 （少） 虛九 〈（少六十七秒）〉 危十五度半室十七壁八 （太） 右北方七宿，九十四度 〈（六十七秒）〉。
Dipper: 25 degrees; Ox: 7 degrees. (small) Girl: 11 degrees. (small) Emptiness: 9 degrees. (67 small seconds) Rooftop: 15½ degrees; Room: 17 degrees; Wall: 8 degrees. (large) At right: the seven northern lodges, 94 degrees. (67 seconds)
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奎十六半婁十二胃十五昴十一 （少） 畢十七 （少） 觜半參十半右西方七宿，八十三度。
Legs: 16½ degrees; Bond: 12 degrees; Stomach: 15 degrees; Hairy Head: 11 degrees. (small) Net: 17 degrees. (small) Turtle Beak: ½ degree; Three Stars: 10½ degrees. At right: the seven western lodges, 83 degrees.
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井三十三 （少） 鬼二半柳十三 （太） 星六 （太） 張十七 （少） 翼十八軫十七右南方七宿，一百九度 （少）。
Well: 33 degrees. (small) Ghost: 2½ degrees; Willow: 13 degrees. (large) Star: 6 degrees. (large) Extended Net: 17 degrees. (small) Wings: 18 degrees; Axletree: 17 degrees. At right: the seven southern lodges, 109 degrees. (small)
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角十二亢九 （少） 氐十六房五 （太） 心六 （少） 尾十九 （少） 箕十半右東方七宿，七十九度。 求冬至赤道日度
Horn: 12 degrees; Neck: 9 degrees. (small) Base: 16 degrees; Room: 5 degrees. (large) Heart: 6 degrees. (small) Tail: 19 degrees. (small) Winnowing Basket: 10½ degrees. At right: the seven eastern lodges, 79 degrees. To find the winter solstice equatorial solar degree.
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置通積分，以周天分去之； 余，日法而一，為度，不滿，退除為分秒； 以百為母，命起赤道虛宿六度外，去之，不滿宿，即得所求年天正冬至加時日躔赤道宿度及分秒。 〈（其在尋斯幹之東西者，先以裏差加減通積分。）〉
Set the universal accumulated parts and divide by the circuit-of-heaven parts; with the remainder, divide by the day unit to obtain degrees, and reduce what does not fill a degree into parts and seconds; with 100 as divisor, count from 6 degrees beyond Emptiness in the equatorial lodges and discard full lodges; the remainder gives the equatorial lodge, degrees, and parts-seconds of the sun at the winter solstice of the sought year with time-of-day. (For places east and west of Xanadu, first apply the inner difference to add or subtract from the universal accumulated parts.)
38
求春分夏至秋分赤道日度
To find the equatorial solar degrees at the spring equinox, summer solstice, and autumn equinox.
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置天正冬至加時赤道日度，累加象限，滿赤道宿次，去之，即各得春分、夏至、秋分加時日在宿度及分秒。 求四正赤道宿積度
Set the equatorial solar degree at the civil-year winter solstice with time-of-day, add the quadrant limit cumulatively, and when it fills the equatorial lodge sequence discard the quotient; the lodge, degrees, and parts-seconds of the sun at the spring equinox, summer solstice, and autumn equinox with time-of-day are thereby obtained. To find the accumulated equatorial lodge degrees after the four cardinal points.
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置四正赤道宿全度，以四正赤道日度及分秒減之，余為距後度； 以赤道宿度累加之，各得四正後赤道宿積度及分秒。 求赤道宿積度入初末限
Set the full degree of the equatorial lodge at each cardinal point, subtract the equatorial solar degree and parts-seconds at that cardinal point, and the remainder is the degrees after the interval; add the equatorial lodge degrees cumulatively to obtain the accumulated equatorial lodge degrees and parts-seconds after each cardinal point. To find whether the accumulated equatorial lodge degrees fall within the beginning or end limit.
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視四正後赤道宿積度及分，在四十五度六十五分五十四秒半以下，為入初限； 以上者，用減象限，余為入末限。 求二十八宿黃道度
Inspect the accumulated equatorial lodge degrees and parts after a cardinal point; if they are 45 degrees 65 parts 54½ seconds or below, they fall within the beginning limit; if above, subtract from the quadrant limit; the remainder falls within the end limit. To find the ecliptic degrees of the twenty-eight lodges.
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置四正後赤道宿入初末限度及分，減一百一度； 余，以初末限度及分乘之，進位，滿百為分，分滿百為度； 至後以減、分後以加赤道宿積度，為其宿黃道積度； 以前宿黃道積度減之， 〈（其四正之宿，先加象限，然後以前宿減之。）〉 為其宿黃道度及分。 〈（其分就近約為太半少。）〉
Set the beginning- or end-limit degrees and parts for the equatorial lodge after a cardinal point and subtract 101 degrees; multiply the remainder by the beginning- or end-limit degrees and parts, carry when needed, and when the product fills 100 convert to parts, and when parts fill 100 convert to degrees; after a solstice subtract from, and after an equinox add to, the accumulated equatorial lodge degrees to obtain the lodge's accumulated ecliptic degrees; subtract the previous lodge's accumulated ecliptic degrees, (For the lodges at the four cardinal points, first add the quadrant limit, then subtract the previous lodge.) to obtain that lodge's ecliptic degrees and parts. (The parts are rounded to the nearest large, half, or small degree.)
43
黃道宿度
Degrees of the ecliptic lodges.
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鬥二十三牛七女十一虛九 〈（少六十七秒）〉 危十六室十八 （少） 壁九半右北方七宿，九十四度 〈（六十七秒）〉。
Dipper: 23 degrees; Ox: 7 degrees; Girl: 11 degrees; Emptiness: 9 degrees. (67 small seconds) Rooftop: 16 degrees; Room: 18 degrees. (small) Wall: 9½ degrees. At right: the seven northern lodges, 94 degrees. (67 seconds)
45
奎十七 （太） 婁十二 （太） 胃十五半昴十一畢十六半觜半參九 （太） 右西方七宿，八十三度 （太）。
Legs: 17 degrees. (large) Bond: 12 degrees. (large) Stomach: 15½ degrees; Hairy Head: 11 degrees; Net: 16½ degrees; Turtle Beak: ½ degree; Three Stars: 9 degrees. (large) At right: the seven western lodges, 83 degrees. (large)
46
井三十半鬼二半柳十三 （少） 星六 （太） 張十七 （太） 翼二十軫十八半右南方七宿，一百九度 （少）。
Well: 30½ degrees; Ghost: 2½ degrees; Willow: 13 degrees. (small) Star: 6 degrees. (large) Extended Net: 17 degrees. (large) Wings: 20 degrees; Axletree: 18½ degrees. At right: the seven southern lodges, 109 degrees. (small)
47
角十二 （太） 亢九 （太） 氐十六 （少） 房五 （太） 心六尾十八 （少） 箕九半右東方七宿，七十八度 （少）。
Horn: 12 degrees. (large) Neck: 9 degrees. (large) Base: 16 degrees. (small) Room: 5 degrees. (large) Heart: 6 degrees; Tail: 18 degrees. (small) Winnowing Basket: 9½ degrees. At right: the seven eastern lodges, 78 degrees. (small)
48
前黃道宿度，依今歷歲差所在算定。 如上考往古，下驗將來，當據歲差，每移一度，依術推變當時宿度，然後可步七曜，知其所在。 求天正冬至加時黃道日度
The ecliptic lodge degrees given above are fixed by calculation according to the annual precession in the present calendar. As when examining antiquity above and verifying the future below, one must rely on annual precession: for each shift of one degree, derive the lodge degrees of that epoch by the prescribed method; only then can the seven luminaries be stepped and their positions known. To find the ecliptic solar degree at the civil-year winter solstice with time-of-day.
49
以冬至加時赤道日度分秒，減一百一度，余以冬至加時赤道日度及分秒乘之，進位，滿百為分，分滿百為度，命曰黃赤道差； 用減冬至加時赤道日度及分秒，即得所求年天正冬至加時黃道日度及分秒。
From the equatorial solar degree and parts-seconds at the winter solstice with time-of-day, subtract 101 degrees; multiply the remainder by that equatorial degree and its parts-seconds, carry when needed, and when the product fills 100 convert to parts and when parts fill 100 convert to degrees—this is the yellow-red difference; Subtract this from the equatorial solar degree and parts-seconds at the winter solstice with time-of-day to obtain the ecliptic solar degree and parts-seconds at the civil-year winter solstice of the sought year with time-of-day.
50
求二十四氣加時黃道日度
To find the ecliptic solar degree with time-of-day for each of the twenty-four solar terms.
51
置所求年冬至日躔黃赤道差，以次年黃赤道差減之，余以所求氣數乘之，二十四而一； 所得，以加其氣中積度及約分，以其氣初日盈縮數盈加縮減之，用加冬至加時黃道日度，依宿次去之，即各得其氣加時黃道日躔宿度及分秒。 〈（如其年冬至加時赤道宿度空分秒在歲差以下者，即加前宿全度，然求黃赤道差，余依術算。）〉
Set the yellow-red difference at the winter solstice sun's position for the sought year, subtract the next year's yellow-red difference, multiply the remainder by the number of the term sought, and divide by twenty-four; Add the result to that term's mid-term accumulated degrees and rounded parts; apply the term's first-day expansion-contraction value—add when in surplus, subtract when in contraction—and add this to the ecliptic solar degree at the winter solstice with time-of-day; discard full lodges in lodge sequence to obtain each term's ecliptic lodge, degrees, and parts-seconds of the sun's motion with time-of-day. (If the fractional parts of the equatorial lodge degree at the winter solstice with time-of-day fall below the annual precession value, add the previous lodge's full degree before computing the yellow-red difference; compute the remainder by the prescribed method.)
52
求二十四氣及每日晨前夜半黃道日度
To find the ecliptic solar degree before dawn and at midnight for each solar term and each day.
53
副置其恒氣小余，以其氣初日損益率乘之， 〈（盈縮之損益。）〉 萬約之，應益者盈加縮減，應損者盈減縮加，其副日法除之，為度，不滿，退除為分秒，以減其氣加時黃道日度，即得其氣初日晨前夜半黃道日度。 每日加一度，以萬乘之，又以每日損益數， 〈（盈縮之損益。）〉 應益者盈加縮減，應損者盈減縮加，為每日晨前夜半黃道日度及分秒。
Set aside the minor remainder of its constant qi and multiply it by that term's first-day increase-decrease rate, (Increase-decrease of expansion and contraction.) Reduce by ten thousand; where the rate is to increase, add when in surplus and subtract when in contraction; where it is to decrease, subtract when in surplus and add when in contraction; divide the auxiliary value by the day unit to obtain degrees, and reduce the remainder into parts and seconds; subtract this from the term's ecliptic solar degree with time-of-day to obtain the ecliptic solar degree before dawn and at midnight on the first day of that term. Add one degree for each day, multiply by ten thousand, and then by each day's increase-decrease number, (Increase-decrease of expansion and contraction.) where the rate is to increase, add when in surplus and subtract when in contraction; where it is to decrease, subtract when in surplus and add when in contraction—to obtain each day's ecliptic solar degree and parts-seconds before dawn and at midnight.
54
求每日午中黃道日度
To find the ecliptic solar degree at noon for each day.
55
置一萬分，以所求入氣日損益數加減， 〈（益者，盈加縮減； 損者，盈減縮加。）〉 半之，滿百為分，不滿為秒，以加其日晨前夜半黃道日度，即其日午中日躔黃道宿度及分秒。
Set 10,000 parts and add or subtract the daily increase-decrease number for the day within the term, (Where increasing, add when in surplus and subtract when in contraction; (where decreasing, subtract when in surplus and add when in contraction.) Halve it; when the result fills 100 convert to parts and what does not fill becomes seconds; add this to that day's ecliptic solar degree before dawn and at midnight to obtain the ecliptic lodge, degrees, and parts-seconds of the sun's motion at noon that day.
56
求每日午中黃道積度
To find the accumulated ecliptic solar degree at noon for each day.
57
以二至加時黃道日度，距至所求日午中黃道日度，為入二至後黃道日積度及分秒。 求每日午中黃道入初末限
From the ecliptic solar degree at a solstice with time-of-day, take the interval to the sought day's ecliptic solar degree at noon as the accumulated ecliptic solar degrees and parts-seconds after entry into the solstice. To find whether each day's noon ecliptic degree falls within the beginning or end limit.
58
視二至後黃道積度，在四十三度一十二分八十七秒之以下為初限； 以上，用減象限，余為入末限。 其積度，滿象限去之，為二分後黃道積度； 在四十八度一十八分二十一秒之以下，為初限； 以上，用減象限，余為入末限。
View the accumulated ecliptic degrees after a solstice: if at or below 43° 12′ 87″, this is within the beginning limit; If above, subtract from the quadrant limit; the remainder falls within the end limit. When the accumulated degrees fill the quadrant limit, discard the quotient to obtain the accumulated ecliptic degrees after an equinox; If at or below 48° 18′ 21″, this is within the beginning limit; If above, subtract from the quadrant limit; the remainder falls within the end limit.
59
求每日午中赤道日度
To find the equatorial solar degree at noon for each day.
60
以所求日午中黃道積度，入至後初限、分後末限度及分秒，進三位，加二十萬二千五十少，開平方除之，所得減去四百四十九半，余在初限者，直以二至赤道日度加而命之； 在末限者，以減象限，余以二分赤道日度加而命之，即每日午中赤道日度。
Take the accumulated ecliptic degree at the sought noon and the degrees and parts-seconds within the beginning limit after a solstice or the end limit after an equinox; advance three places, add 202,050 plus a fractional small amount, extract the square root and divide; subtract 449½ from the result; if within the beginning limit, add directly to the equatorial solar degree at the solstice and count off in lodges; If within the end limit, subtract from the quadrant limit and add the remainder to the equatorial solar degree at an equinox and count in lodges to obtain the equatorial solar degree at each noon.
61
以所求日午中黃道積度，入至後末限、分後初限度及分秒，進三位，用減三十萬三千五十少，開平方除之，所得，以減五百五十半，其在初限者，以所減之余，直以二分赤道日度加而命之； 在末限者，以減象限，余以二至赤道日度加而命之，即每日午中赤道日度。
Take the accumulated ecliptic degree at the sought noon and the degrees and parts-seconds within the end limit after a solstice or the beginning limit after an equinox; advance three places, subtract from 303,050 plus a fractional small amount, extract the square root and divide; subtract 550½ from the result; if within the beginning limit, add the remainder after subtraction directly to the equatorial solar degree at an equinox and count off in lodges; If within the end limit, subtract from the quadrant limit and add the remainder to the equatorial solar degree at a solstice and count in lodges to obtain the equatorial solar degree at each noon.
62
太陽黃道十二次入宮宿度危十三度三十九分五十九秒外入衛分陬訾之次，辰在亥。 奎二度三十五分八十五秒外入魯分降婁之次，辰在戌。 胃四度二十四分三十三秒外入趙分大梁之次，辰在酉。
Ecliptic palace-entry degrees for the sun's twelve regions: beyond 13° 39′ 59″ in Rooftop the sun enters Wei's territory in Zouzi, with the celestial mark at hai. Beyond 2° 35′ 85″ in Legs the sun enters Lu's territory in Jianglou, with the celestial mark at xu. Beyond 4° 24′ 33″ in Stomach the sun enters Zhao's territory in Daliang, with the celestial mark at you.
63
畢七度九十六分二十秒外入晉分實沈之次，辰在申。 井九度四十七分一十秒外入秦分鶉首之次，辰在未。 柳四度九十五分二十六秒外入周分鶉火之次，辰在午。
Beyond 7° 96′ 20″ in Net the sun enters Jin's territory in Shichen, with the celestial mark at shen. Beyond 9° 47′ 10″ in Well the sun enters Qin's territory in Chunshou, with the celestial mark at wei. Beyond 4° 95′ 26″ in Willow the sun enters Zhou's territory in Chunhuo, with the celestial mark at wu.
64
張十五度五十六分三十五秒外入楚分鶉尾之次，辰在巳。 軫十度四十四分五秒外入鄭分壽星之次，辰在辰。 氐一度七十七分七十七秒外入宋分大火之次，辰在卯。
Beyond 15° 56′ 35″ in Extended Net the sun enters Chu's territory in Chunwei, with the celestial mark at si. Beyond 10° 44′ 5″ in Axletree the sun enters Zheng's territory in Shouxing, with the celestial mark at chen. Beyond 1° 77′ 77″ in Base the sun enters Song's territory in Great Fire, with the celestial mark at mao.
65
尾三度九十七分七十二秒外入燕分析木之次，辰在寅。 鬥四度三十六分六十六秒外入吳越分星紀之次，辰在醜。 女二度九十一分九十一秒外入齊分玄枵之次，辰在子。
Beyond 3° 97′ 72″ in Tail the sun enters Yan's territory in Splintered Wood, with the celestial mark at yin. Beyond 4° 36′ 66″ in Dipper the sun enters Wu-Yue's territory in Star Chronicle, with the celestial mark at chou. Beyond 2° 91′ 91″ in Girl the sun enters Qi's territory in Dark Emblem, with the celestial mark at zi.
66
求入宮時刻
To find the time of palace entry.
67
各置入宮宿度及分秒，以其日晨前夜半日度減之， （相近一度之間者求之。 余以日法乘其分， 〈（其秒從於下，亦通乘之。）〉 為實； 以其日太陽行分為法； 實如法而一，所得，依發斂加時求之，即得其日太陽入宮時刻及分秒。
For each, set the palace-entry lodge degrees and parts-seconds and subtract that day's solar degree before dawn and at midnight, (Compute when within about one degree. Multiply the remainder's parts by the day unit, (Carry the seconds down and multiply them through as well.) for the dividend; use that day's solar motion parts as divisor; Divide the dividend by the divisor; obtain the quotient and find the time by the issuing-and-gathering procedure to obtain that day's time and parts-seconds of the sun's palace entry.
68
步晷漏術中限，一百八十二日六十二分一十八秒。 冬至初限、夏至末限，六十二日二十分。 夏至初限、冬至末限，一百二十日四十二分。
Stepwise computation: gnomon and water-clock. Mid-limit: 182 days 62 parts 18 seconds. Winter solstice beginning-limit and summer solstice end-limit: 62 days 20 parts. Summer solstice beginning-limit and winter solstice end-limit: 120 days 42 parts.
69
冬至永安晷影常數，一丈二尺八寸三分。 夏至永安晷影常數，一尺五寸六分。 周法，一千四百二十八。 內外法，一萬八百九十六。
Yong'an winter-solstice gnomon-shadow constant: 1 zhang 2 chi 8 cun 3 fen. Yong'an summer-solstice gnomon-shadow constant: 1 chi 5 cun 6 fen. Zhou divisor: 1428. Inner-outer divisor: 10,896.
70
半法，二千六百一十五。 日法四分之三，三千九百二十二半。 日法四分之一，一千三百七半。 昏明分，一百三十分七十五秒。
Half-divisor: 2615. Three-fourths of the day unit: 3922½. One-fourth of the day unit: 1307½. Twilight parts: 130 parts 75 seconds.
71
昏明刻，二刻一百五十六分九十秒。 刻法，三百一十三分八十秒。 秒母，一百。 求午中入氣中積
Twilight notches: 2 notches 156 parts 90 seconds. Quarter-hour divisor: 313 parts 80 seconds. Second divisor: 100. To find the accumulated mid-term interval at noon within a qi.
72
置所求日大余及半法，以所入氣大小余減之，為其日午中入氣； 以加其氣中積，為其日午中中積。 〈（小余以日法除，為約分。）〉 求二至後午中入初末限
Set the major remainder of the sought day and the half-divisor, subtract the major and minor remainders of the qi entered, and obtain that day's noon interval within the qi; Add this to the mid-term accumulation of that qi to obtain that day's noon mid-accumulation. (Divide the minor remainder by the day unit to obtain reduced parts.) To find whether noon after the two solstices falls within the beginning or end limit.
73
置午中中積及分，如中限以下，為冬至後； 以上，去中限，為夏至後。 其二至後，如在初限以下，為初限； 以上，覆減中限，余為入末限也。 求午中晷影定數
Set the noon mid-accumulation and its parts; if equal to or below the mid-limit, it is after the winter solstice; if above, subtract the mid-limit to obtain the interval after the summer solstice. Within each post-solstice period, if equal to or below the beginning-limit, it falls in the beginning-limit; if above, subtract the mid-limit in reverse, and the remainder is the entry into the end-limit. To find the fixed noon gnomon shadow length.
74
視冬至後初限、夏至後末限，百通日內分，自相乘，副置之，以一千四百五十除之； 所得，加五萬三百八，折半限分並之，除其副為分，分滿十為寸，寸滿十為尺，用減冬至地中晷影常數，為所求晷影定數。
For the beginning-limit after winter solstice and the end-limit after summer solstice, multiply the day's inner parts by one hundred and square them; set aside the product and divide by 1450; Add 50,308 to the result and combine with the halved limit parts; divide the assistant product to obtain parts, convert ten parts to one cun and ten cun to one chi, subtract from the central-earth winter-solstice gnomon-shadow constant, and the fixed gnomon shadow length sought is obtained.
75
視夏至後初限、冬至後末限，百通日內分，自相乘，為上位； 下置入限分，以二百二十五乘之，百約之，加一十九萬八千七十五，為法； 〈（夏至前後半限以上者，減去半限，列於上位，下置半限，各百通日內分，先相減，後相乘，以七千七百除之，所得以加其法。）〉 及除上位為分，分滿十為寸，寸滿十為尺，用加夏至地中晷影常數，為所求晷影定數。
For the beginning-limit after summer solstice and the end-limit after winter solstice, multiply the day's inner parts by one hundred and square them for the upper term; Set the entry-limit parts below, multiply by 225 and reduce by one hundred, add 198,075 to obtain the divisor; (On either side of summer solstice when above the half-limit, subtract the half-limit and place the result in the upper position; set the half-limit below; for each, multiply inner parts by one hundred, subtract first and multiply second, divide by 7700, and add the result to the divisor.) Divide the upper term by the divisor to obtain parts, convert ten parts to one cun and ten cun to one chi, add to the central-earth summer-solstice gnomon-shadow constant, and the fixed gnomon shadow length sought is obtained.
76
求四方所在晷影
To find the gnomon shadow at each quarter of the realm.
77
各於其處測冬夏二至晷數，乃相減之，余為其處二至晷差； 亦以地中二至晷數相減，為地中二至晷差。 其所求日在冬至後初限、夏至後末限者，如在半限以下，倍之； 半限以上，覆減全限，余亦倍之； 並入限日，三因，折半，以日為分，十分為寸，以減地中二至晷差，為法； 置地中冬至晷影常數，以所求日地中晷影定數減之，余以其處二至晷差乘之，為實； 實如法而一，所得，以減其處冬至晷數，即得其處其日晷影定數。 所求日在夏至後初限、冬至後末限者，如在半限以下，倍之； 半限以上，覆減全限，余亦倍之； 並入限日，三因，四除，以日為分，十分為寸，以加地中二至晷差，為法； 置所求日地中晷影定數，以地中夏至晷影常數減之，余以其處二至晷差乘之，為實； 實如法而一，所得，以加其處夏至晷數，即得其處其日晷影定數。
At each place measure the gnomon lengths at the winter and summer solstices and subtract them; the remainder is that place's solstitial gnomon difference; Subtract the central-earth solstitial gnomon lengths likewise to obtain the central-earth solstitial gnomon difference. When the sought day falls in the beginning-limit after winter solstice or the end-limit after summer solstice, if equal to or below the half-limit, double it; if above the half-limit, subtract the full limit in reverse and double the remainder as well; Add the days within the limit, multiply by three, and halve; take days as parts and ten parts as one cun, subtract from the central-earth solstitial gnomon difference to obtain the divisor; Set the central-earth winter-solstice gnomon-shadow constant, subtract the fixed central-earth gnomon shadow for the sought day, and multiply the remainder by that place's solstitial gnomon difference for the dividend; Divide the dividend by the divisor; subtract the result from that place's winter-solstice gnomon length to obtain the fixed gnomon shadow for that place on that day. When the sought day falls in the beginning-limit after summer solstice or the end-limit after winter solstice, if equal to or below the half-limit, double it; if above the half-limit, subtract the full limit in reverse and double the remainder as well; Add the days within the limit, multiply by three, and divide by four; take days as parts and ten parts as one cun, add to the central-earth solstitial gnomon difference to obtain the divisor; Set the fixed central-earth gnomon shadow for the sought day, subtract the central-earth summer-solstice gnomon-shadow constant, and multiply the remainder by that place's solstitial gnomon difference for the dividend; Divide the dividend by the divisor; add the result to that place's summer-solstice gnomon length to obtain the fixed gnomon shadow for that place on that day.
78
二十四氣陟降及日出分 〈（以下表格略）〉 二分前後陟降率
Ascent-descent and sunrise parts for the twenty-four solar terms. (The tables below are omitted.) Ascent-descent rates before and after the equinoxes.
79
春分前三日，太陽入赤道內，秋分後三日，太陽出赤道外，故其陟降與他日不倫，今各別立數而用之。
Three days before the spring equinox the sun enters within the equator, and three days after the autumn equinox it exits beyond the equator; therefore its ascent and descent do not match other days, and separate numbers are now established and used for each.
80
驚蟄，十二日陟四。 〈（六十七、一十六。）〉 此為末率，於此用畢。 〈（其減差亦止於此也。）〉
Awakening of Insects: on the twelfth day, ascend four. (67, 16.) This is the terminal rate; its use ends here. (The subtractive correction also stops here.)
81
十三日陟四。 〈（四十一、六。）〉 十四日陟四。 〈（三十八、九十。）〉 十五日陟四。
On the thirteenth day, ascend four. (41, 6.) On the fourteenth day, ascend four. (38, 90.) On the fifteenth day, ascend four.
82
秋分，初日降四。 〈（三十八。）〉 一日降四。 〈（三十九。）〉 二日降四。 〈（五十九）〉。 三日降四。 〈（六十八。）〉 此為初率，始用之。 〈（其加差亦始於此也。）〉
Autumn Equinox: on the first day, descend four. (38.) On the first day, descend four. (39.) On the second day, descend four. (59.) On the third day, descend four. (68.) This is the initial rate; its use begins here. (The additive correction also begins here.)
83
求每日日出入晨昏半晝分
To find each day's sunrise, sunset, dawn, dusk, and half-day parts.
84
各以陟降初率，陟減降加其氣初日日出分，為一日下日出分； 以增損差 〈（仍加減加減差。）〉 增損陟降率，馴積而加減之，即為每日日出分； 覆減日法，余為日入分； 以日出分減日入分，半之，為半晝分； 以昏明分減日出分，為晨分； 加日入分，為昏分。
For each term, apply the initial ascent-descent rate—subtract when ascending and add when descending—to the first day's sunrise parts of that qi to obtain the sunrise parts below day one; with the increase-decrease difference (still apply the additive-subtractive correction.) Increase or decrease the ascent-descent rates and apply them in sequential accumulation to obtain each day's sunrise parts; subtract in reverse from the day unit to obtain the sunset parts; subtract sunrise parts from sunset parts and halve the result for the half-day parts; subtract twilight parts from sunrise parts to obtain dawn parts; add sunset parts to obtain dusk parts.
85
求日出入辰刻
To find the double-hours and quarter-hours of sunrise and sunset.
86
置日出入分，以六因之，滿辰法而一，為辰數； 不盡，刻法除之，為刻，不滿為分。 命子正算外，即得所求。 求晝夜刻
Set the sunrise-sunset parts, multiply by six, and divide by the double-hour divisor to obtain the double-hour count; with the remainder, divide by the quarter-hour divisor to obtain quarter-hours, and what does not fill becomes parts. Count from midnight beyond the counting rod to obtain the result sought. To find the day and night quarter-hours.
87
置日出分，十二乘之，刻法而一，為刻，不滿為分，即為夜刻； 覆減一百，余為晝刻及分秒。 求更點率置晨分，四因之，退位，為更率； 二因更率，退位，為點率。
Set the sunrise parts, multiply by twelve, and divide by the quarter-hour divisor to obtain quarter-hours and parts as the night quarter-hours; subtract in reverse from one hundred; the remainder is the day quarter-hours and parts-seconds. To find watch and point rates: set the dawn parts, multiply by four, and shift one place to obtain the watch rate; Multiply the watch rate by two and shift one place to obtain the point rate.
88
求更點所在辰刻
To find the double-hour and quarter-hour of a watch or point.
89
置更點率，以所求更點數因之，又六因之，內加更籌刻，滿辰法而一，為辰數； 不盡，滿刻法，除之，為刻數； 不滿，為分； 命其日辰刻算外，即得所求。
Set the watch-point rate, multiply by the sought number of watches or points and again by six, add the watch tally quarter-hours within, and divide by the double-hour divisor to obtain the double-hour count; with the remainder, divide by the quarter-hour divisor to obtain the quarter-hour count; what does not fill becomes parts; count from that day's double-hour and quarter-hour beyond the counting rod to obtain the result sought.
90
求四方所在漏刻
To find the clepsydra quarter-hours at each quarter of the realm.
91
各於所在下水漏，以定其處冬至或夏至夜刻，乃與五十刻相減，余為至差刻。 置所求日黃道去赤道內外度及分，以至差刻乘之，進一位，如二百三十九而一，為刻； 不盡，以刻法乘之，退除為分； 內減外加五十刻，即得所求日夜刻； 以減百刻，余為晝刻。 〈（其日出入辰刻及更點差率等，並依前術求之。）〉
At each place set the water clepsydra running and fix the winter- or summer-solstice night quarter-hours there; subtract from fifty quarter-hours, and the remainder is the solstitial difference in quarter-hours. Set the ecliptic's degrees and parts north or south of the equator for the sought day, multiply by the solstitial-difference quarter-hours, advance one place, and divide by 239 to obtain quarter-hours; with the remainder, multiply by the quarter-hour divisor and divide down to obtain parts; subtract within and add without to fifty quarter-hours to obtain the sought day and night quarter-hours; subtract from one hundred quarter-hours; the remainder is the day quarter-hours. (The double-hours and quarter-hours of sunrise and sunset, watch and point difference rates, and the like are all found by the methods above.)
92
求黃道內外度
To find the ecliptic degrees north or south of the equator.
93
置日出之分，如日法四分之一以上，去之，余為外分； 如日法四分之一以下，覆減之，余為內分。 置內外分，千乘之，如內外法而一，為度，不滿，退除為分秒，即為黃道去赤道內外度； 內減外加象限，即得黃道去極度。
Set the sunrise parts; if they reach one-quarter of the day unit or more, discard that amount; the remainder is the outer parts; If below one-quarter of the day unit, subtract in reverse; the remainder is the inner parts. Set the inner or outer parts, multiply by one thousand, and divide by the inner-outer divisor to obtain degrees; with the remainder, divide down for parts and seconds; this is the ecliptic's degrees north or south of the equator; Subtract within and add without to the quadrant limit to obtain the ecliptic's degrees from the pole.
94
求距中度及更差度
To find the degrees from the meridian and the watch-interval difference.
95
置半法，以晨分減之，余為距中分； 百乘之，如周法而一，為距中度； 用減一百八十三度一十二分八十三秒半，余四因，退位，為每更差度。 求昏明五更中星
Set the half-divisor, subtract the dawn parts, and the remainder is the distance-from-mid parts; Multiply by one hundred and divide by the Zhou divisor to obtain the degrees from the meridian; Subtract this from 183 degrees 12 parts 83½ seconds, multiply the remainder by four, shift one place, and obtain the watch-interval difference per watch. To find the culminating stars at dusk, dawn, and the five watches.
96
置距中度，以其日午中赤道日度加而命之，即昏中星所格宿次，因為初更中星； 以更差度累加之，滿赤道宿次，去之，即得逐更及明中星。 步月離術
Set the degrees from the meridian, add that day's noon equatorial solar degree and count in lodges; the lodge reached is the culminating star at dusk and the first watch; Add the watch-interval difference cumulatively; when it fills the equatorial lodge sequence, discard the quotient; the culminating stars for each successive watch and at dawn are thereby obtained. Procedure for lunar motion.
97
轉終分，一十四萬四千一百一十，秒六千二十，微六十。 轉終日，二十七，余二千九百，秒六千二十，微六十。 轉中日，一十三，余四千六十五，秒三千一十，微三十。
Revolution cycle parts: 144,110, seconds 6,020, micro-units 60. Revolution cycle days: 27, remainder 2,900, seconds 6,020, micro-units 60. Revolution mid-cycle days: 13, remainder 4,065, seconds 3,010, micro-units 30.
98
朔差日，一，余五千一百四，秒三千九百七十九，微四十。 象策，七，余二千一，秒二千五百。 秒母，一萬。 微母，一百。
New-moon difference: 1 day, remainder 5,104 parts, seconds 3,979, micro-units 40. Quarter-moon divisor: 7 days, remainder 2,001 parts, seconds 2,500. Second divisor: 10,000. Micro divisor: 100.
99
上弦度，九十一，分三十一，秒四十一 （太）。 望度，一百八十二，分六十二，秒八十三半。 下弦度，二百七十三，分九十四，秒二十五 （少）。 月平行度，十三，分三十六，秒八十七半。
First-quarter arc: 91 degrees 31 parts 41 seconds (large) Full-moon arc: 182 degrees 62 parts 83½ seconds. Last-quarter arc: 273 degrees 94 parts 25 seconds (small) Moon mean daily motion: 13 degrees 36 parts 87½ seconds.
100
分秒母，一百。 七日初數，四千六百四十八，末數，五百八十二。 十四日初數，四千六十五，末數，一千一百六十五。 二十一日初數，三千四百八十三，末數，一千七百四十七。
Parts-seconds divisor: 100. Seven-day initial count: 4,648; final count: 582. Fourteen-day initial count: 4,065; final count: 1,165. Twenty-one-day initial count: 3,483; final count: 1,747.
101
二十八日初數，二千九百一。 求經朔弦望入轉 〈（凡稱秒者，微從之，他仿此。）〉
Twenty-eight-day initial count: 2,901. To find the mean new moon, quarter moon, and full moon at their entry into revolution. (Whenever seconds are given, micro-units follow; other cases follow this rule.)
102
置天正朔積分，以轉終分及秒去之，不盡，如日法而一，為日，不滿為余秒，即天正十一月經朔入轉日及余秒； 以象策累加之，去命如前，得弦望經日加時入轉及余秒； 徑求次朔入轉，即以朔差加之。 〈（加減裏差，即得中朔弦望入轉及余秒。）〉
Set the civil-year new-moon accumulated parts, cast out the revolution cycle parts and seconds, divide the remainder by the day unit to obtain days, with the unfilled portion as remainder-seconds; this yields the day and remainder-seconds of entry into revolution for the mean new moon of the eleventh month of the civil year; Add the quarter-moon divisor cumulatively, cast out and name as before, and obtain the day-and-hour entry into revolution with remainder-seconds for each quarter and full moon; To find directly the next new moon's entry into revolution, add the new-moon difference. (Apply the inner difference to add or subtract, and obtain the entry into revolution with remainder-seconds for the mid-term new moon, quarter moon, and full moon.)
103
求轉定分及積度朓朒。 〈（以下表格略）〉 求中朔弦望入轉朓朒定數
To find the revolution fixed parts and accumulated waning-waxing degrees. (Table below omitted.) To find the fixed waning-waxing number for the mid-term new moon, quarter moon, and full moon at entry into revolution.
104
置入轉小余，以其日算外損益率乘之，如日法而一，所得，以損益朓朒積，為定數。 其四七日下余，如初數以下，初率乘之，如初數而一，以損益朓朒積，為定數； 如初數以上，以初數減之，余乘末率，如末數而一，用減初率，余如朓朒積，為定數。 其十四日下余，如初數以上，以初數減之，余乘末率，如末數而一，為朓朒定數。
Set the entry-into-revolution minor remainder, multiply by the deficit-surplus rate beyond the count for that day, and divide by the day unit; apply the result to the waning-waxing accumulation to obtain the fixed number. On the fourth and seventh days and below in remainder: if below the initial count, multiply by the initial rate and divide by the initial count, and apply the result to the waning-waxing accumulation for the fixed number; If at or above the initial count, subtract the initial count; multiply the remainder by the final rate and divide by the final count, subtract from the initial rate, and take the remainder as the waning-waxing accumulation for the fixed number. On the fourteenth day and below in remainder: if at or above the initial count, subtract the initial count; multiply the remainder by the final rate and divide by the final count to obtain the waning-waxing fixed number.
105
求朔弦望中日
To find the mid-day of the new moon, quarter moon, and full moon.
106
以尋斯幹城為準，置相去地裏，以四千三百五十九乘之，退位，萬約為分，曰裏差； 以加減經朔弦望小余，滿與不足，進退大余，即中朔弦望日及余。 〈（以東加之，以西減之。）〉
Taking Xanadu as the standard, set the distance in li, multiply by 4,359, shift one place, and reduce by ten thousand to obtain parts, called the inner difference; Add or subtract this from the mean new moon, quarter moon, and full moon minor remainders; on fullness or shortfall, advance or retreat the major remainder to obtain the mid-term new moon, quarter moon, and full moon day and remainder. (Add for places to the east and subtract for places to the west.)
107
求朔弦望定日
To find the fixed new moon, quarter moon, and full moon days.
108
置中朔弦望小余，朓減朒加入氣入轉朓朒定數，滿與不足，進退大余，命壬戌算外，各得定朔弦望日辰及余。 定朔幹名與後朔同者，其月大； 不同者，其月小； 月內無中氣者，為閏。 視定朔小余，秋分後在日法四分之三以上者，進一日； 春分後，定朔日出分與春分日出分相減之，余者，三約之，用減四分之三； 定朔小余及此分以上者，亦進一日； 或有交，虧初於日入前者，不進之。 定弦望小余，在日出分以下者，退一日； 或有交，虧初於日出前者，小余雖在日出後，亦退之。 如望在十七日者，又視定朔小余在四分之三以下之數， 〈（春分後用減定之數。）〉 與定望小余在日出分以上之數相校之，朔少望多者，望不退，而朔猶進之； 望少朔多者，朔不進，而望猶退之。 〈（日月之行，有盈縮遲疾； 加減之數，或有四大三小。 若循常當察加時早晚，隨所近而進退之，使不過四大三小。）〉
Set the mid-term new moon, quarter moon, and full moon minor remainders, subtract waning and add waxing, then add the entry-into-qi and entry-into-revolution waning-waxing fixed number; on fullness or shortfall, advance or retreat the major remainder and count from renxu beyond the counting rod; each fixed new moon, quarter moon, and full moon with its day, double-hour, and remainder is thereby obtained. When the stem of the fixed new moon matches that of the following new moon, the month is long; when they differ, the month is short; a month with no mid-qi within it is intercalary. Inspect the fixed new moon minor remainder: after the autumn equinox, if it reaches three-quarters of the day unit or above, advance one day; After the spring equinox, subtract the spring-equinox sunrise parts from the fixed-new-moon sunrise parts, divide the remainder by three, and subtract that from three-quarters of the day unit; if the fixed new moon minor remainder reaches this number of parts or above, advance one day as well; if there is a crossing and first contact comes before sunset, do not advance. When the fixed quarter or full moon minor remainder falls below the sunrise parts, retreat one day; if there is a crossing and first contact comes before sunrise, retreat even when the minor remainder lies after sunrise. For a full moon on the seventeenth day, also inspect whether the fixed new moon minor remainder lies at or below three-quarters, (After the spring equinox apply the subtracted fixed number.) compare it with the fixed full moon minor remainder at or above the sunrise parts; if the new moon is less and the full moon greater, do not retreat the full moon but still advance the new moon; if the full moon is less and the new moon greater, do not advance the new moon but still retreat the full moon. (The motions of the sun and moon have expansion and contraction, fast and slow; and the numbers added or subtracted may yield four long months and three short. (When proceeding routinely one should inspect whether the time-of-day is early or late and advance or retreat according to what is nearer, so as not to exceed four long and three short.)
109
求定朔弦望中積
To find the mid-accumulation for the fixed new moon, quarter moon, and full moon.
110
置定朔弦望小余，與中朔弦望小余相減之，余以加減經朔弦望入氣日余， 〈（中朔弦望，少即加之，多即減之。）〉 即為定朔弦望入氣； 以加其氣中積，即為定朔弦望中積。 〈（其餘，以日法退除為分秒。）〉
Set the fixed new moon, quarter moon, and full moon minor remainders and subtract the mid-term new moon, quarter moon, and full moon minor remainders; use the remainder to add or subtract from the mean day-remainder of entry into a qi for the new moon, quarter moon, and full moon, (for the mid-term new moon, quarter moon, and full moon, add if less and subtract if greater.) this is the fixed new moon, quarter moon, and full moon at their entry into a qi; add this to the mid-term accumulation of that qi to obtain the mid-accumulation for the fixed new moon, quarter moon, and full moon. (with the remainder, divide down by the day unit to obtain parts and seconds.)
111
求定朔弦望加時日度
To find the solar degrees at the time-of-day for the fixed new moon, quarter moon, and full moon.
112
置定朔弦望約余，以所入氣日損益率乘之， 〈（盈縮之損益。）〉 萬約之，以損益其下盈縮積，乃盈加縮減定朔弦望中積，又以冬至加時日躔黃道宿度加之，依宿次去之，即得定朔弦望加時日所在度分秒。
Set the reduced remainder for the fixed new moon, quarter moon, and full moon and multiply by the increase-decrease rate for the day of the qi entered, (the increase-decrease of expansion and contraction.) reduce by ten thousand and apply to the expansion-contraction accumulation below; then add when in expansion and subtract when in contraction from the mid-accumulation of the fixed new moon, quarter moon, and full moon; add the ecliptic lodge and degree of the sun's motion at the winter solstice with time-of-day; discard full lodges; the degrees, parts, and seconds of the sun at the time-of-day for each fixed new moon, quarter moon, and full moon are thereby obtained.
113
又法：置定朔弦望約余，副之，以乘其日盈縮之損益率，萬約之，應益者盈加縮減，應損者盈減縮加，其副滿百為分，分滿百為度，以加其日夜半日度，命之，各得其日加時日躔黃道宿次。 〈（若先於歷中註定每日夜半日度，即用此法為妙也。）〉
Alternate method: set the reduced remainder for the fixed new moon, quarter moon, and full moon and set aside a copy; multiply by that day's increase-decrease rate for expansion and contraction and reduce by ten thousand; when increase is called for, add in expansion and subtract in contraction, and when decrease is called for, subtract in expansion and add in contraction; when the copy fills 100 convert to parts and when parts fill 100 convert to degrees; add to that day's midnight solar degree and count in lodges; each yields the ecliptic lodge of the sun's motion at the time-of-day for that day. (If each day's midnight solar degree has already been entered in the calendar, this method is especially effective.)
114
求定朔弦望加時月度
To find the lunar degrees at the time-of-day for the fixed new moon, quarter moon, and full moon.
115
凡合朔加時日月同度，其定朔加時黃道日度即為定朔加時黃道月度； 弦望，各以弦望度加定朔弦望加時黃道日度，依宿次去之，即得定朔弦望加時黃道月度及分秒。
Whenever at conjunction the sun and moon share the same degree at the time-of-day, the ecliptic solar degree at the fixed new moon with time-of-day is the ecliptic lunar degree at the fixed new moon with time-of-day; For the quarter and full moons, add the quarter-moon or full-moon arc to the ecliptic solar degree at the fixed event with time-of-day and discard full lodges; the ecliptic lunar degree with parts and seconds at the fixed new moon, quarter moon, and full moon with time-of-day is thereby obtained.
116
求夜半午中入轉
To find entry into revolution at midnight and noon.
117
置中朔入轉，以中朔小余減之，為中朔夜半入轉。 又中朔小余，與半法相減之，余以加減中朔加時入轉， 〈（中朔少如半法，加之； 多如半法，減之。）〉 為中朔午中入轉。 若定朔大余有進退者，亦加減轉日，否則因中為定，每日累加一日，滿轉終日及余秒，去命如前，各得每日夜半午中入轉。 〈（求夜半，因定朔夜半入轉累加之； 求午中，因定朔午中入轉累加之； 求加時入轉者，如求加時入氣之術法。）〉
Set the mid-term new moon's entry into revolution and subtract the mid-term new moon minor remainder to obtain entry into revolution at midnight for the mid-term new moon. Subtract the mid-term new moon minor remainder from the half-divisor and use the remainder to add or subtract from the mid-term new moon's entry into revolution at the time-of-day, (if the mid-term new moon remainder is less than the half-divisor, add; (if greater than the half-divisor, subtract.) this is entry into revolution at noon for the mid-term new moon. If the fixed new moon major remainder has been advanced or retreated, adjust the revolution day as well; otherwise take the mid-term value as fixed; add one day cumulatively each day; when it fills the revolution cycle day and remainder-seconds, cast out and name as before; entry into revolution at midnight and noon for each day is thereby obtained. (To find midnight, accumulate entry into revolution from the fixed new moon at midnight; to find noon, accumulate from the fixed new moon entry into revolution at noon; (to find entry into revolution at the time-of-day, use the method for finding entry into qi at the time-of-day.)
118
求加時及夜半月度
To find the lunar degrees at the time-of-day and at midnight.
119
置其日入轉算外轉定分，以定朔弦望小余乘之，如日法而一，為加時轉分； 〈（分滿百為度。）〉 減定朔弦望加時月度，為夜半月度。 以相次轉定分累加之，即得每日夜半月度。 〈（或朔至弦望，或至後朔，皆可累加之。 然近則差少，遠則差多。 置所求前後夜半相距月度為行度，計其日相距入轉積度，與行度相減，余以相距日數除之，為日差行度。 多日差加每日轉定分行度，少日差減每日轉定分而用之可也。 欲求速，即用此數。 欲究其微，而可用後術。）〉
Set that day's entry into revolution beyond the count and its fixed revolution parts, multiply by the fixed new moon, quarter moon, or full moon minor remainder, divide by the day unit, and obtain the revolution parts at the time-of-day; (when parts fill one hundred, they become degrees.) Subtract the fixed-event lunar degree at the time-of-day to obtain the midnight lunar degree. Add the successive fixed revolution parts cumulatively to obtain the midnight lunar degree for each day. (Whether from new moon to quarter or full moon, or to the following new moon, one may accumulate; but when near the difference is small and when far it is large. Set the lunar degrees between the sought midnight and the preceding or following midnight as the motion arc, compute the accumulated entry-into-revolution degrees between those days, subtract the motion arc, divide the remainder by the number of intervening days, and obtain the daily motion difference. When the day-difference is large, add the daily motion from fixed revolution parts; when small, subtract the daily fixed revolution parts; that suffices. For speed, use this figure. (For finer detail, use the procedure below.)
120
求晨昏月度
To find the lunar degrees at dawn and dusk.
121
置其日晨分，乘其日算外轉定分，日法而一，為晨轉分； 用減轉定分，余為昏轉分。 又以朔望定小余，乘轉定分，日法而一，為加時分，以減晨昏轉分，為前； 不足，覆減之，為後； 乃前加後減加時月度，即晨昏月度所在宿度及分秒。
Set that day's dawn parts, multiply by that day's fixed revolution parts beyond the count, divide by the day unit, and obtain the dawn revolution parts; subtract these from the fixed revolution parts; the remainder is the dusk revolution parts. Multiply the fixed revolution parts by the fixed minor remainder of the new or full moon, divide by the day unit to obtain the time-of-day parts, subtract from the dawn or dusk revolution parts for the before case; if insufficient, subtract in reverse for the after case; then add before and subtract after from the lunar degree at the time-of-day to obtain the lodge, degrees, parts, and seconds of the dawn or dusk lunar position.
122
求朔弦望晨昏定程
To find the fixed dawn and dusk intervals for the new moon, quarter moon, and full moon.
123
各以其朔昏定月減上弦昏定月，余為朔後昏定程。 以上弦昏定月，減望昏定月，余為上弦後昏定程。 以望晨定月，減下弦晨定月，余為望後晨定程。 以下弦晨定月，減後朔晨定月，余為下弦後晨定程。
For each, subtract the first-quarter dusk fixed moon from the new-moon dusk fixed moon; the remainder is the fixed dusk interval after the new moon. Subtract the full-moon dusk fixed moon from the first-quarter dusk fixed moon; the remainder is the fixed dusk interval after the first quarter. Subtract the last-quarter dawn fixed moon from the full-moon dawn fixed moon; the remainder is the fixed dawn interval after the full moon. Subtract the next new moon's dawn fixed moon from the last-quarter dawn fixed moon; the remainder is the fixed dawn interval after the last quarter.
124
求每日轉定度
To find the daily fixed revolution degrees.
125
累計每定程相距日下轉積度，與晨昏定程相減，余以相距日數除之，為日差； 〈（定程多，加之； 定程少，減之。）〉 以加減每日轉定分，為轉定度； 因朔弦望晨昏月，每日累加之，滿宿次去之，為每日晨昏月度及分秒。 〈（凡註歷，朔日已後註昏月，望後一日註晨月。）〉 古歷有九道月度，其數雖繁，亦難削去，具其術如後。
Accumulate the entry-into-revolution degrees under each fixed interval between the days, subtract the dawn or dusk fixed interval, divide the remainder by the number of intervening days, and obtain the day-difference; (if the fixed interval is greater, add; (if less, subtract.) add or subtract this from each day's fixed revolution parts to obtain the fixed revolution degrees; Starting from the dawn or dusk moon at the new moon, quarter moon, or full moon, add cumulatively each day, cast out full lodges, and obtain the dawn or dusk lunar degree with parts and seconds for each day. (In calendar annotation, record the dusk moon from the new-moon day onward and the dawn moon from the day after the full moon.) Earlier calendars included nine-path lunar monthly degrees; though the figures are numerous and hard to omit, the procedure is set forth below.
126
求正交日辰
To find the day and double-hour of true crossing.
127
置交終日及余秒，以其月經朔加時入交泛日及余秒減之，余為平交入其月經朔加時後日算及余秒； 〈（中朔同。）〉 以加其月中朔大小余，其大余命壬戌算外，即得平交日辰及余秒。 〈（求次交者，以交終日及余秒加之，如大余滿紀法，去之，命如前，即得次平交日辰及余秒也。）〉
Set the crossing cycle day and remainder-seconds and subtract the general day and remainder-seconds of entry into crossing at the month's mean new moon with time-of-day; the remainder is the day count and remainder-seconds after the mean new moon with time-of-day at which mean crossing occurs; (the same for the mid-term new moon.) Add the month's mid-term new moon major and minor remainders; count the major remainder from renxu beyond the counting rod to obtain the day, double-hour, and remainder-seconds of mean crossing. (To find the next crossing, add the crossing cycle day and remainder-seconds; if the major remainder fills the era unit, cast it out and count as before to obtain the next mean-crossing day, double-hour, and remainder-seconds.)
128
求平交入轉朓朒定數
To find the waning-waxing fixed number for mean crossing's entry into revolution.
129
置平交小余，加其日夜半入轉，余以乘其日損益率，日法而一，所得，以損益其日下朓朒積，為定數。 求平交日辰
Set the mean-crossing minor remainder, add midnight entry into revolution for that day, multiply the remainder by that day's increase-decrease rate, divide by the day unit, and apply the result to increase or decrease the waning-waxing accumulation below that day to obtain the fixed number. To find the day and double-hour of mean crossing.
130
置平交小余，以平交入轉朓朒定數朓減朒加之，滿與不足，進退日辰，即得正交日辰及余秒； 與定朔日辰相距，即得所在月日。 求中朔加時中積
Set the mean-crossing minor remainder, apply the waning-waxing fixed number for mean crossing's entry into revolution (subtract when waning, add when waxing), and on fullness or shortfall advance or retreat the day and double-hour to obtain the true-crossing day, double-hour, and remainder-seconds; compare with the fixed new moon day and double-hour to obtain the month and day. To find the mid-accumulation at the mid-term new moon with time-of-day.
131
各以其月中朔加時入氣日及余，加其氣中積及余，其日命為度，其餘，以日法退除為分秒，即其月中朔加時中積度及分秒。 求正交加時黃道月度
For each, add the day and remainder of entry into a qi at the month's mid-term new moon with time-of-day to the mid-accumulation and remainder of that qi; count the days as degrees, divide the remainder down by the day unit into parts and seconds, and obtain the mid-accumulation in degrees, parts, and seconds at the mid-term new moon with time-of-day. To find the ecliptic lunar degree at true crossing with time-of-day.
132
置平交入中朔加時後日算及余秒，以日法通日內余進二位，如三萬九千一百二十一為度，不滿，退除為分秒，以加其月中朔加時中積，然後以冬至加時黃道日度加而命之，即得其月正交加時月離黃道宿度及分秒。 如求次交者，以交中度及分秒加而命之，即得所求。
Set the day count and remainder-seconds after mean crossing at the mid-term new moon with time-of-day, convert the remainder within the day through the day unit advancing two places, take 39,121 as the divisor for degrees, divide down the remainder into parts and seconds, add to the mid-accumulation at the mid-term new moon with time-of-day, then add and count from the ecliptic solar degree at the winter solstice with time-of-day to obtain the ecliptic lodge, degree, parts, and seconds of the moon's departure at true crossing with time-of-day for that month. To find the next crossing, add the crossing mid-arc in degrees, parts, and seconds and count in lodges to obtain the result.
133
求黃道宿積度
To find the accumulated ecliptic lodge degrees.
134
置正交加時黃道宿全度，以正交加時月離黃道宿度及分秒減之，余為距後度及分秒； 以黃道宿度累加之，即各得正交後黃道宿積度及分秒。 求黃道宿積度入初末限
Set the full ecliptic lodge degree at true crossing with time-of-day and subtract the ecliptic lodge degree, parts, and seconds of the moon's departure; the remainder is the degrees and parts-seconds after the distance; add the ecliptic lodge degrees cumulatively to obtain the accumulated ecliptic lodge degrees and parts-seconds after each true crossing. To find entry into the initial or final limit for accumulated ecliptic lodge degrees.
135
置黃道宿積度及分秒，滿交象度及分秒去之，余在半交象以下為初限； 以上者，減交象度，余為末限。 〈（入交積度、交象度，並在《交會篇》中。）〉
Set the accumulated ecliptic lodge degrees and parts-seconds, cast out full crossing-image degrees and parts-seconds; if the remainder is at or below half the crossing image, it is the initial limit; if above, subtract the crossing-image degrees; the remainder is the final limit. (Entry-crossing accumulation degrees and crossing-image degrees are both in the Crossing Conjunctions chapter.)
136
求月行九道宿度
To find the lodge degrees of the moon's nine paths.
137
凡月行所交，冬入陰歷，夏入陽歷，月行青道； 〈（冬至夏至後，青道半交在春分之宿，當黃道東； 立冬立夏後，青道半交在立春之宿，當黃道東南； 至所沖之宿，亦皆如之也。 宜細推。）〉 冬入陽歷，夏入陰歷，月行白道； 〈（冬至夏至後，白道半交在秋分之宿，當黃道西； 立冬立夏後，白道半交在立秋之宿，當黃道西北； 至所沖之宿，亦如之也。）〉 春入陽歷，秋入陰歷，月行朱道； 〈（春分秋分後，朱道半交在夏至之宿，當黃道南； 立春立秋後，朱道半交在立夏之宿，當黃道西南； 至所沖之宿，亦如之也。）〉 春入陰歷，秋入陽歷，月行黑道。 〈（春分秋分後，黑道半交在冬至之宿，當黃道北； 立春立秋後，黑道半交在立冬之宿，當黃道東北； 至所沖之宿，亦如之也。）〉 四時離為八節，至陰陽之所交，皆與黃道相會，故月行有九道。 各以所入初末限度及分，減一百一度，余以所入初入初末限度及分乘之，半而退位為分，分滿百為度，命為月道與黃道泛差。
Whenever the moon crosses, in winter it enters the yin calendar and in summer the yang calendar, and the moon moves on the green path; (After the winter and summer solstices, the green path's half-crossing lies at the lodge of the spring equinox, east of the ecliptic; after Start of Winter and Start of Summer, the green path's half-crossing lies at the lodge of Start of Spring, southeast of the ecliptic; at the opposed lodge it is likewise. (This should be worked out in detail.) In winter it enters the yang calendar and in summer the yin calendar, and the moon moves on the white path; (After the winter and summer solstices, the white path's half-crossing lies at the lodge of the autumn equinox, west of the ecliptic; after Start of Winter and Start of Summer, the white path's half-crossing lies at the lodge of Start of Autumn, northwest of the ecliptic; (at the opposed lodge it is likewise.) In spring it enters the yang calendar and in autumn the yin calendar, and the moon moves on the vermilion path; (After the spring and autumn equinoxes, the vermilion path's half-crossing lies at the lodge of the summer solstice, south of the ecliptic; after Start of Spring and Start of Autumn, the vermilion path's half-crossing lies at the lodge of Start of Summer, southwest of the ecliptic; (at the opposed lodge it is likewise.) In spring it enters the yin calendar and in autumn the yang calendar, and the moon moves on the black path. (After the spring and autumn equinoxes, the black path's half-crossing lies at the lodge of the winter solstice, north of the ecliptic; after Start of Spring and Start of Autumn, the black path's half-crossing lies at the lodge of Start of Winter, northeast of the ecliptic; (at the opposed lodge it is likewise.) The four seasons divide into eight nodes; where yin and yang cross, all meet the ecliptic; hence the moon has nine paths. For each, take the entered initial or final limit in degrees and parts, subtract 101 degrees, multiply the remainder by the degrees and parts of entry into the initial limit within the initial or final limit, halve and shift one place for parts, convert parts filling 100 to degrees, and name the result the general difference between the lunar path and the ecliptic.
138
凡日以赤道內為陰，外為陽； 月以黃道內為陰，外為陽。 故月行正交，入夏至後宿度內為同名，入冬至後宿度內為異名。 其在同名者，置月行與黃道泛差，九因之，八約之，為定差； 半交後，正交前，以差減； 正交後，半交前，以差加； 〈（此加減出入六度，異如黃赤道相交同名之差，若較之漸異，則隨交所在遷變不常。）〉 仍以正交度距秋分度數，乘定差，如象限而一，所得，為月道與赤道定差； 前加者為減，減者為加。 其在異名者，置月行與黃道泛差，七因之，八約之，為定差； 半交後，正交前，以差加； 正交後，半交前，以差減； 〈（此加減出入六度，正如黃赤道相交異名之差，若較之漸同，則隨交所在遷變不常。）〉 仍以正交度距春分度數，乘定差，如象限而一，所得，為月道與赤道定差； 前加者為減，減者為加，各加減黃道宿積度，為九道宿積度； 以前宿九道積度減之，為其宿九道度及分秒。 〈（其分就近約為太、半、少，論春夏秋冬，以四時日所在宿度為正。）〉
For the sun, within the equator is yin and without is yang; for the moon, within the ecliptic is yin and without is yang. Therefore at true lunar crossing, degrees within lodges after the summer solstice are same-name and those after the winter solstice are different-name. In the same-name case, set the general difference between the lunar motion and the ecliptic, multiply by nine and reduce by eight to obtain the fixed difference; after half-crossing and before true crossing, subtract the difference; after true crossing and before half-crossing, add the difference; (This addition and subtraction varies by six degrees, unlike the same-name difference at ecliptic-equator crossing; compared gradually it diverges and shifts with the crossing location.) Still multiply the fixed difference by the true-crossing degrees' distance from the autumn equinox and divide by the quadrant to obtain the fixed difference between the lunar path and the equator; what was added before becomes subtraction and what was subtracted becomes addition. In the different-name case, set the general difference between the lunar motion and the ecliptic, multiply by seven and reduce by eight to obtain the fixed difference; after half-crossing and before true crossing, add the difference; after true crossing and before half-crossing, subtract the difference; (This addition and subtraction of six degrees in and out is just like the difference of unlike names at the yellow-red intersection; if compared they gradually coincide, the shift varies irregularly with the location of the node.) Again multiply the fixed difference by the true-crossing degrees' distance from the spring equinox and divide by the quadrant limit; the result is the fixed difference between the lunar path and the equator; what was previously added becomes subtraction and what was subtracted becomes addition; add or subtract each from the accumulated ecliptic lodge degrees to obtain the accumulated nine-path lodge degrees; subtract the previous lodge's accumulated nine-path degrees to obtain that lodge's nine-path degrees and parts-seconds. (The parts are rounded to the nearest large, half, or small degree; for spring, summer, autumn, and winter, the lodge degrees where the sun stands in each season serve as the standard.)
139
求正交加時月離九道宿度
To find the nine-path lodge degree of the moon's departure at true conjunction with time-of-day.
140
以正交加時黃道日度及分，減一百一度，余以正交度及分乘之，半而退位為分，分滿百為度，命為月道與黃道泛差。 其在同名者，置月行與黃道泛差，九因之，八約之，為定差，以加； 仍以正交度距秋分度數乘定差，如象限而一，所得，為月道與赤道定差，以減。 其異名者，置月行與黃道泛差，七因之，八約之，為定差，以減； 仍以正交度距春分度數，乘定差，如象限而一，所得，為月道與赤道定差，以加。 置正交加時黃道月度及分，以二差加減之，即為正交加時月離九道宿度及分。
Subtract 101 degrees from the ecliptic solar degree and parts at true conjunction with time-of-day; multiply the remainder by the true-crossing degrees and parts, halve and shift down for parts, and when parts fill 100 convert to degrees; this is called the general difference between the lunar path and the ecliptic. When of like name, set the lunar motion and ecliptic general difference, multiply by nine and reduce by eight to obtain the fixed difference, and add it; Again multiply the fixed difference by the true-crossing degrees' distance from the autumn equinox and divide by the quadrant limit; the result is the fixed difference between the lunar path and the equator, and subtract it. When of unlike name, set the lunar motion and ecliptic general difference, multiply by seven and reduce by eight to obtain the fixed difference, and subtract it; Again multiply the fixed difference by the true-crossing degrees' distance from the spring equinox and divide by the quadrant limit; the result is the fixed difference between the lunar path and the equator, and add it. Set the ecliptic lunar degree and parts at true conjunction with time-of-day, add or subtract the two differences, and the nine-path lodge degree and parts of the moon's departure at true conjunction with time-of-day are obtained.
141
求定朔弦望加時月所在度
To find the degree occupied by the moon at the true new moon, quarter moon, and full moon with time-of-day.
142
置定朔加時日躔黃道宿次，凡合朔加時，月行潛在日下，與太陽同度，是為加時月離宿次； 各以弦望度及分秒，加其所當弦望加時日躔黃道宿度，滿宿次，去之，命如前，各得定朔弦望加時月所在黃道宿度及分秒。
Set the ecliptic lodge position of the sun at the true new moon with time-of-day; whenever it is conjunction with time-of-day, the moon moves hidden beneath the sun at the same degree, and this is the lodge position of the moon's departure at that hour; For each, add the quarter- or full-moon degrees and parts-seconds to the ecliptic solar degree at the corresponding quarter or full moon with time-of-day, discard full lodges as before, and the ecliptic lodge, degrees, and parts-seconds occupied by the moon at each true new moon, quarter moon, and full moon with time-of-day are thereby obtained.
143
求定朔弦望加時九道月度
To find the nine-path lunar degree at the true new moon, quarter moon, and full moon with time-of-day.
144
各以定朔弦望加時月離黃道宿度及分秒，加前宿正交後黃道積度，為定朔弦望加時正交後黃道積度； 如前求九道積度，以前宿九道積度減之，余為定朔弦望加時九道月離宿度及分秒。 〈（其合朔加時，若非正交，則日在黃道，月在九道，所入宿度雖多少不同，考其兩極若繩準。 故云月行潛在日下，與太陽同度，即為加時。 九道月度，求其晨昏夜半月度，並依前術。）〉
For each, add the moon's departure ecliptic lodge degrees and parts-seconds at the true new moon, quarter moon, or full moon with time-of-day to the accumulated ecliptic degrees after true crossing at the previous lodge, obtaining the accumulated ecliptic degrees after true crossing at that hour; as before, find the nine-path accumulated degrees and subtract the previous lodge's accumulated nine-path degrees; the remainder is the nine-path lodge, degrees, and parts-seconds of the moon's departure at the true new moon, quarter moon, or full moon with time-of-day. (At conjunction with time-of-day, if it is not true crossing, the sun is on the ecliptic and the moon on the nine paths; the lodges entered may differ in degree, but examine the two poles as if by a plumb line. Hence it is said that the moon moves hidden beneath the sun at the same degree, and that is the time-of-day. (For nine-path lunar degrees, find the lunar degrees at dawn, dusk, and midnight by the same methods as above.)